21 problems

Let $K_5$ be the complete graph on five vertices. Complete-graph $K_5$ conjecture. … The source presents this as a conjectural computational pattern and gives no resolution.

Let $K_4$ be the complete graph on four vertices. Complete-graph $K_4$ conjecture. For every $m\geq4$, … The paper reports verification for $m\leq14$, but does not establish the st…

Let $P_3|P_3$ denote the graph formed by gluing two triangles along an edge, and let $H^{i,j}_{{\mathcal A}_m}(G)$ denote the corresponding graph cohomology. Square-with-diagonal c…

For $n>4$, let $W_n$ be the wheel graph, let $W_n^{out}$ be obtained by deleting an edge from its polygonal part, and let $W_n^{in}$ be obtained by deleting a spike edge. Wheel and…

Let $K_n$ be the complete graph on $n$ vertices, and let $w^1_{{\mathcal A}_3}(K_n)$ denote the width of the torsion in $H^{1,*}_{{\mathcal A}_3}(K_n)$. Width conjecture for comple…

Let $W_n$ be the wheel graph, the cone over an $(n-1)$-gon, and let $w^1_{{\mathcal A}_3}(W_n)$ denote the width of the torsion in $H^{1,*}_{{\mathcal A}_3}(W_n)$. Width conjecture…

Let $G_{t,s^k}$ be the graph obtained by gluing a triangle and $k$ squares in a sequence along edges, and let $w^1_{{\mathcal A}_3}(G)$ denote the width of the torsion in…

Let $G_{t,s^k}$ be the graph obtained by gluing a triangle and $k$ squares in a sequence along edges, with $v=3+2k$ vertices. Torsion formula for $G_{t,s^k}$. … This is one of the…

Let $P_{t,k}=P_3|P_k$ be the graph with $k+1$ vertices obtained by gluing a triangle and a $k$-gon along an edge. Let $w^1_{{\mathcal A}_3}(G)$ be the width of the torsion in…

Let $G$ be a loopless graph, and let $H^{i,*}_{{\mathcal A}_m}(G)$ denote its graph cohomology associated with ${\mathcal A}_m$. A triangle is a cycle of length $3$, and a square i…

Let ${\mathcal A}_m$ be the algebra used to define graph cohomology, let $P_v$ be the polygon with $v$ vertices, and let $H^{i,j}_{{\mathcal A}_m}(P_v)$ denote its bigraded cohomol…

Let ${\mathcal A}_m$ be the algebra used to define graph cohomology, let $P_v$ be the polygon with $v$ vertices, and let $H^{i,j}_{{\mathcal A}_m}(P_v)$ denote its bigraded cohomol…

Let $G$ be a graph, and let $H^{**}_{{\mathcal A}_m}(G)$ denote its graph cohomology associated with the algebra ${\mathcal A}_m$. A graph is loopless if it has no loops, and a cyc…

Let $W_{n,k}^\ast$ be the Kontsevich cycle indexed by nonnegative integers $n$ and $k$, and let $\widetilde{\kappa}_j$ denote the corresponding modified kappa class. Write…

Nontriviality conjecture. For every $k=1,2,\dots$, the element $X_{4k+2}$ defines a non-trivial cohomology class in

Let $\Omega^{\bullet}_{\mathrm{can}}$ be the graded space of canonical differential forms, let $\mathbb{L}(\Omega^{\bullet}_{\mathrm{can}})$ be the free Lie algebra on it, and let…

A finite graph $\Gamma$ is called naturally oriented if $H^1(\Gamma)$ has a basis whose supports are isomorphic, in the orientation-preserving sense, to cycles whose edges all poin…

Let $P_m$ be the $m$-prism with ladder perfect matching $L$, and let $H^{i,j}(P_m,L)$ denote the associated bigraded cohomology. Prism graph cohomology conjecture. If $(P_m,L)$ is…

Let $\mathrm{GC}_3$ be the graph complex, and let $L_k$ denote its loop graphs. The partner classes are those classes killed by the loop graphs in the spectral sequence described i…

Let $H(\mathrm{GC}_2)$ denote the cohomology of the even graph complex. Concentration conjecture. The even graph cohomology is concentrated in cohomological degrees…

Let $\mathrm{GC}_3$ be the odd graph complex, let $H(\mathrm{GC}_3)$ denote its cohomology, and let $L_{4k-1}$ be the loop graphs of the indicated loop order. The source refers to…